# Application of angular velocity to Euler angles

According to a post here Angular Velocity expressed via Euler Angles you can express angular velocity from euler angles. If I choose Y-Z-Y as a rotation sequence the expression becomes.

$$\theta_r, \theta_p, \theta_y$$ = roll, pitch, yaw

$$\vec{\omega} = \dot{\theta_r} \hat{y} + R_z(\theta_p)( \left( \dot{\theta_p} \hat{z} + R_y(\theta_y) \left( \dot{\theta_y} \hat{y} \right) \right)$$

which becomes

according to this

where

which does not make sense.

does it make sense and does it still work in this case?

• With Y-Z-Y rotations the angular velocity vector is $$\vec{\omega} = \dot{\theta_r} \hat{y} + R_y(\theta_r) \left( \dot{\theta_p} \hat{z} + R_z(\theta_p) \left( \dot{\theta_y} \hat{y} \right) \right)$$ I think you misunderstood my answer. Commented Jul 22, 2019 at 1:02
• @ja72 Yeah I did but you misunderstood the order. it is first roll, then pitch, then yaw. Thus the angular velocity becomes $$\vec{\omega} = \dot{\theta_y}\hat{y}+R_y(\theta_y)\left(\dot{\theta_p}\hat{z}+R_z(\theta_p)\left(\dot{\theta_r}\hat{y}\right)\right)$$ which finally evaluates to $$\begin{bmatrix} \omega_x \\ \omega_y \\ \omega_z \end{bmatrix} = \begin{bmatrix} a\dot{\theta_p} - bc\dot{\theta_r} \\ \dot{\theta_y}+d\dot{\theta_r} \\b\dot{\theta_p} + ac\dot{\theta_r} \end{bmatrix}$$ is this correct? thanks for giving me your time. Commented Jul 23, 2019 at 0:08
• Indeed, but my answer below has the correct order. I don't know what the coefficients $a$, $b$, $c$ ... are so I don't know about your result. Please review my answer and award it if it was helpful. Commented Jul 23, 2019 at 1:42
• @ja72 the coefficients are defined in the main post quite clearly and your answer below has the angles in the wrong order. I already said this. When rotating you first rotate in the Y axis with $\theta_r$ then in Z with $\theta_p$ lastly Y again with $\theta_y$. You wrote them in the reverse order even in your answer. Commented Jul 23, 2019 at 1:47

Suppose you have a Y-Z-Y scheme with a corresponding sequence of rotation angles $$\theta_y$$, $$\theta_p$$ and $$\theta_r$$.

After the first rotation (yaw), the 3×3 orientation matrix $$\mathrm{E}_y$$ and angular velocity vector $$\vec{\omega}_y$$ is

\begin{aligned} \mathrm{E}_y & = \mathrm{rot}(\hat{j}, \theta_y) & \vec{\omega}_y & = \dot{\theta}_y \left(\hat{j}\right) \end{aligned} \;\tag{1}

The above should be self-evident. Now consider the second rotation and the orientation matrix $$\mathrm{E}_p$$ and angular velocity vector $$\vec{\omega}_p$$. Since the local axes are rotated by the first rotation we have

\begin{aligned} \mathrm{E}_p & = \mathrm{E}_y \mathrm{rot}(\hat{k}, \theta_p) & \vec{\omega}_p & = \dot{\theta}_y \left( \hat{j} \right) + \dot{\theta}_p \left( \mathrm{E}_y \hat{k} \right) \end{aligned} \;\tag{2}

Finally, with the third rotation we extend this pattern to find the final orientation matrix $$\mathrm{E}$$ and the final rotation velocity vector $$\vec{\omega}$$

\begin{aligned} \mathrm{E} & = \mathrm{E}_p \mathrm{rot}(\hat{j}, \theta_r) & \vec{\omega} & = \dot{\theta}_y \left( \hat{j} \right) + \dot{\theta}_p \left( \mathrm{E}_y \hat{k} \right) + \dot{\theta}_r \left( \mathrm{E}_p \hat{j} \right) \end{aligned} \;\tag{3}

The last part is re-written as

\begin{aligned} \mathrm{E} & =\mathrm{rot}(\hat{j}, \theta_y)\mathrm{rot}(\hat{k}, \theta_p) \mathrm{rot}(\hat{j}, \theta_r) & \vec{\omega} & = \dot{\theta}_y \hat{j} + \mathrm{rot}(\hat{j}, \theta_y) \left( \hat{k} \dot{\theta}_p + \mathrm{rot}(\hat{k}, \theta_p) \hat{j} \dot{\theta}_r \right) \end{aligned} \;\tag{4}

This expands out to the following jacobian formulation

$$\vec{\omega} = \begin{bmatrix} 0 & \sin(\theta_y) & -\sin(\theta_p)\cos(\theta_y) \\ 1 & 0 & \cos(\theta_p) \\ 0 & \cos(\theta_y) & \sin(\theta_p) \sin(\theta_y) \end{bmatrix} \pmatrix{ \dot{\theta}_y \\ \dot{\theta}_p \\ \dot{\theta}_r } \;\tag{5}$$

• Proof of the above comes from expanding the identity $\dot{\mathrm{E}} = \vec{\omega} \times \mathrm{E}$ and the derivative product rule applied to the sequence of rotations in $\mathrm{E}$. Commented Jul 22, 2019 at 1:38
• @fullnitrus - Equation (4) above is identical to your equation you posted in the comments. Commented Jul 23, 2019 at 12:45
• I checked, equation (5) is identical to $$\begin{bmatrix} \omega_x \\ \omega_y \\ \omega_z \end{bmatrix} = \begin{bmatrix} a\dot{\theta_p} - bc\dot{\theta_r} \\ \dot{\theta_y}+d\dot{\theta_r} \\b\dot{\theta_p} + ac\dot{\theta_r} \end{bmatrix}$$ Commented Jul 23, 2019 at 12:54
• "After the first rotation (yaw)" Commented Jul 23, 2019 at 15:04
• I call it the first rotation because it is always about a fixed axis (the vertical). I come from robotics where kinematics are recursive from the base to the tip, and this is built in a similar fashion. I guess you would consider first the roll, then the pitch and the yaw, but for me it is logically arranged yaw-pitch-roll. The math is the same though. Commented Jul 23, 2019 at 15:28